Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Braking Distance The grade \(x\) of a hill is a measure of its steepness. For example, if a road rises 10 feet for every 100 feet of horizontal distance, then it has an uphill grade of \(x=\frac{10}{100},\) or \(10 \% .\) The braking (or stopping) distance \(D\) for a car traveling at 50 mph on a wet, uphill grade is given by $$ D(x)=\frac{2500}{30(0.3+x)} $$ (a) Evaluate \(D(0.05)\) and interpret the result. (b) Describe what happens to braking distance as the hill becomes steeper. (c) Estimate the grade associated with a braking distance of 220 feet.

Step-by-Step Solution

Answer

(a) D(0.05) ≈ 238.1 feet; (b) braking distance decreases; (c) grade ≈ 18.48%.

Step 1: Evaluate D(0.05)

We are given the function \( D(x) = \frac{2500}{30(0.3 + x)} \). To find \( D(0.05) \), we substitute \( x = 0.05 \) into the function. So we have:\[D(0.05) = \frac{2500}{30(0.3+0.05)} = \frac{2500}{30 \times 0.35} = \frac{2500}{10.5} \]Calculating this gives:\(D(0.05) \approx 238.1\) feet. This means that on a hill with a grade of 5%, the braking distance is approximately 238.1 feet.

Step 2: Describe Effect of Steeper Hills

To understand what happens to the braking distance as the hill becomes steeper, observe the expression \( D(x) = \frac{2500}{30(0.3 + x)} \). As \( x \) increases, representing a steeper hill, the denominator \( 30(0.3 + x) \) also increases, which causes the entire fraction to decrease. Thus, the braking distance \( D(x) \) decreases as the hill becomes steeper.

Step 3: Estimate Grade for D = 220 Feet

To estimate the grade \( x \) when \( D(x) = 220 \), set\[\frac{2500}{30(0.3 + x)} = 220\]Cross-multiply to solve for \( x \):\[2500 = 220 \times 30(0.3 + x)\]This simplifies to:\[2500 = 660 \times (0.3 + x)\]Divide both sides by 660:\[\frac{2500}{660} = 0.3 + x\]Thus,\[x = \frac{2500}{660} - 0.3 \approx 0.4848 - 0.3 = 0.1848\]The grade associated with a braking distance of 220 feet is approximately 18.48%.

Key Concepts

Grade of a Hill

Understanding the grade of a hill is essential when evaluating how steep a road is. This concept can be particularly important for drivers, construction engineers, and even cyclists. The grade itself is a measure of the incline of the road and is usually expressed as a percentage. For instance, if a road ascends 10 feet vertically for every 100 feet of horizontal distance covered, the grade is calculated as:\[x = \frac{10}{100} = 0.1 = 10\%\]Grades matter because they affect driving conditions, especially braking distances. A higher grade implies a steeper road, which can affect vehicle control and fuel efficiency.

• Roads with higher grades are often more challenging, as they require more effort to ascend and more caution when descending.
• Steep grades can also lead to increased wear on a vehicle’s brakes and engine.

From a safety perspective, understanding the grade allows drivers to anticipate how their vehicle will behave, especially under wet or slippery conditions.

Uphill Braking

Uphill braking is a scenario that many drivers may encounter, especially when driving through hilly terrain. When a car goes uphill, gravity works against its motion, which means the vehicle might slow down naturally. However, knowing the braking distance on different grades remains crucial, particularly under wet or adverse conditions.For the braking distance function given in the exercise:\[D(x) = \frac{2500}{30(0.3 + x)}\]The variable \(x\) represents the grade of the hill. As \(x\) increases, indicating a steeper hill, the braking distance decreases. This is because the car is fighting against the force of gravity more effectively, which aids in stopping. This can be a bit counterintuitive, as one might think uphill increases stopping distance due to the climb, but in terms of physics, more gravitational resistance aids deceleration.

• Always consider the slope and weather conditions when estimating brake performance.
• Higher slopes improve braking on a vehicle by helping slow it down naturally.
• Safer stopping distances on uphill slopes can provide a margin of error in emergency braking.

Stopping Distance Calculation

Stopping distance is a crucial safety metric for vehicular travel, as it represents the total length required to bring a vehicle to a complete stop. Many factors can alter stopping distance, such as speed, road conditions, vehicle weight, and incline. The given formula highlights an uphill scenario, where the braking distance is calculated by the function:\[D(x) = \frac{2500}{30(0.3 + x)}\]Here, some essential points about calculating stopping distance are:

• Speed: Higher speeds generally mean longer stopping distances due to increased kinetic energy.
• Road Surface: Wet or icy roads can significantly extend stopping distances, making calculation adjustments vital.
• Incline: An uphill incline can decrease stopping distances, as gravity assists in slowing the vehicle down.

In the exercise, solving for a specific stopping distance involves algebraic manipulation:- Rearrange the formula to solve for the unknown grade \(x\).- Understand how each factor, including incline percentage, impacts your results.By carefully considering these elements, drivers can better predict and adjust to stopping requirements, enhancing safety on the road.